This work has two aims. The first is to develop a Lagrange–Hamilton framework for the analysis of multi-degree-of-freedom nonlinear systems in which non-conservative effects are included in the variational principle of least action from the outset. The framework is a generalization of the Bateman approach in which a set of adjoint coordinates is introduced. A function termed the M-function is introduced as the Fourier transform over the momenta of the joint probability density function (JPDF) of the displacements and momenta, and it is shown that for statistical systems, this function can be written as an expectation involving the new principle function and a general dimensional constant ℏ. This leads to a concise derivation of the Fokker–Planck–Kolmogorov equation. It is found that the equation governing the M-function can be expressed in terms of the new Hamiltonian by replacing momenta by differential operators, meaning that the function satisfies the same equation as the quantum wave function. This gives rise to the second aim of this work: to explore relations between the developed classical framework and quantum mechanics. It is shown that for an undamped linear system, the solution of the M-function equation yields the response JPDF as a sum of Wigner functions. This classical analysis leads to a number of well-known results from quantum mechanics as ℏ → 0, and the extension of this result to nonlinear systems is discussed. The quantum wave function associated with the Hamiltonian is then considered, and the relevance of this function to the physical system is discussed.
Selection of coordinates for use in evaluating the Sommerfield quantum integrals.---It is well known that these coordinates need not necessarily satisfy Lagrange's equations, and a general criterion for the choice of such variables is given in the present paper, in which it is shown that all sets of
p's and
q's which (a) satisfy Hamilton's equations, (b) separate the variables, and (c) are of such a character that the average value of the kinetic energy is equal to that of $\frac{1}{2}\ensuremath{\Sigma}{1}^{n}{p}_{i}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{q}}_{i}$, will give the correct energy levels when employed in the quantum integrals $\ensuremath{\int}{p}_{i}d{q}_{i}={n}_{i}h$; ($i=1,2,\ensuremath{\cdots}\ifmmode\cdot\else\textperiodcentered\fi{}\ifmmode\cdot\else\textperiodcentered\fi{},n$). This general class of coordinate systems includes as special cases the familiar normalized Schwarzschild angle variables and Lagrangian generalized coordinates which separate the variables.Combination of Hamilton's principle with Trkal's method for quantizing conditionally periodic systems is proposed which, like Hamilton's principle, determines the character of the motion permitted by the Newtonian dynamics, and in addition quantizes the orbits by specializing the constants of integration in accordance with the conventional quantum conditions. This variation principle includes formally in a single equation the results of classical dynamics and the Sommerfield quantum conditions, which together determine the sizes and shapes of electronic orbits. The principle is illustrated by an application to a non-linear oscillator in which the potential involves the second and fourth powers of the displacement from the equilibrium position.